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Convexity and Related Classic Inequalities

In: Thinking in Problems

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  • Alexander A. Roytvarf

Abstract

Concepts of convexity and related techniques are widely used in various branches of mathematics and in applications such as functional analysis, calculus of variations, and mathematical physics, geometry, number theory, theory of games, optimization and other computational methods and convex programming, integral geometry (tomography), and many others. In this chapter we will focus mainly on the basic elements of the theory of convex functions, which can be taught to readers with limited experience. The remaining problems are directed toward specific applications such as convex and linear programming, hierarchy of power means, geometric inequalities, the Hölder and Minkowski inequalities, Young’s inequality and the Legendre transform, and functions on metric and normed vector spaces. The necessary introductory material is provided in these sections. A specially added section discusses the widely known Cauchy-Schwarz-Bunyakovskii (CSB) inequality and Lagrange-type identities in connection with the hierarchy of the power means.

Suggested Citation

  • Alexander A. Roytvarf, 2013. "Convexity and Related Classic Inequalities," Springer Books, in: Thinking in Problems, edition 127, pages 139-195, Springer.
    • Handle: RePEc:spr:sprchp:978-0-8176-8406-8_9
    DOI: 10.1007/978-0-8176-8406-8_9
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